The document discusses solving equations involving squares and square roots. It provides examples of solving equations with squares and square roots, including solving for unknown variables and checking solutions. Key steps shown include taking the square root of both sides of an equation, squaring terms, and isolating the variable. Applications mentioned include physics, engineering, and mechanics.

1. SECTION 3-8
Equations with Squares and Square Roots

2. ESSENTIAL QUESTIONS
• How do you solve problems involving squares?
• How do you solve problems involving square roots?
• Where you’ll see this:
• Physics, safety, engineering, mechanics

3. VOCABULARY
1. Inverse of an Operation:

4. VOCABULARY
1. Inverse of an Operation: The opposite of an operation

5. VOCABULARY
1. Inverse of an Operation: The opposite of an operation
Addition and subtraction

6. VOCABULARY
1. Inverse of an Operation: The opposite of an operation
Addition and subtraction
Multiplication and division

7. QUESTION
What is the opposite of squaring?

8. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9

9. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9
2 4
x =±
9

10. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9
2 4
x =±
9
2
x=±
3

11. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9 +225 +225
2 4
x =±
9
2
x=±
3

12. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9 +225 +225
2
2 4 x = 225
x =±
9
2
x=±
3

13. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9 +225 +225
2
2 4 x = 225
x =±
9 2
x = ± 225
2
x=±
3

14. EXAMPLE 1
Solve each equation. Check the solution.
2 4
a. x = 2
b. x − 225 = 0
9 +225 +225
2
2 4 x = 225
x =±
9 2
x = ± 225
2 x = ±15
x=±
3

15. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v

16. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1

17. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
3 x =2

18. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
3 x =2
3 3

19. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
3 x =2
3 3
2
x=
3

20. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
3 x =2
3 3
2
x=
3
2
2  2
( x) = 
 3

21. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
3 x =2
3 3
2
x=
3
2
2  2 4
( x) = 
 3
x=
9

22. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
2
± 24 = v
3 x =2
3 3
2
x=
3
2
2  2 4
( x) = 
 3
x=
9

23. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
2
± 24 = v
3 x =2
3 3 v = ± 24
2
x=
3
2
2  2 4
( x) = 
 3
x=
9

24. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
2
± 24 = v
3 x =2
3 3 v = ± 24
2
x= or
3
2
2  2 4
( x) = 
 3
x=
9

25. EXAMPLE 1
Solve each equation. Check the solution.
2
c. 3 x +1= 3 d. 24 = v
−1 −1
2
± 24 = v
3 x =2
3 3 v = ± 24
2
x= or
3
2  2
2
4 v ≈ ±4.898979486
( x) = 
 3
x=
9

26. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3

27. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3
2
2  2
( c) = 
 3

28. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3
2
2  2
( c) = 
 3
4
c=
9

29. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3 +10 +10
2
2  2
( c) = 
 3
4
c=
9

30. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3 +10 +10
2
2  2 7w =14
( c) = 
 3
4
c=
9

31. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3 +10 +10
2
2  2 7w =14
( c) = 
 3 2
( 7w ) =14 2
4
c=
9

32. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3 +10 +10
2
2  2 7w =14
( c) = 
 3 2
( 7w ) =14 2
4
c= 7w =196
9

33. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3 +10 +10
2
2  2 7w =14
( c) = 
 3 2
( 7w ) =14 2
4
c= 7w =196
9 7 7

34. EXAMPLE 1
Solve each equation. Check the solution.
2
e. c = f. 7w −10 = 4
3 +10 +10
2
2  2 7w =14
( c) = 
 3 2
( 7w ) =14 2
4
c= 7w =196
9 7 7
w = 28

35. EXAMPLE 2
The velocity v of a satellite moving in a circular orbit near the
surface of Earth is given by the formula v = gr ,
where g represents the force of gravity and r represents the
radius of Earth. Given that g = 9.8 m/sec2 and v = 7.91X103
m/sec, determine the radius of Earth to the nearest meter.

36. EXAMPLE 2
The velocity v of a satellite moving in a circular orbit near the
surface of Earth is given by the formula v = gr ,
where g represents the force of gravity and r represents the
radius of Earth. Given that g = 9.8 m/sec2 and v = 7.91X103
m/sec, determine the radius of Earth to the nearest meter.
v = gr

37. EXAMPLE 2
The velocity v of a satellite moving in a circular orbit near the
surface of Earth is given by the formula v = gr ,
where g represents the force of gravity and r represents the
radius of Earth. Given that g = 9.8 m/sec2 and v = 7.91X103
m/sec, determine the radius of Earth to the nearest meter.
v = gr
3
7.91×10 = 9.8r

38. EXAMPLE 2
3
7.91×10 = 9.8r

39. EXAMPLE 2
3
7.91×10 = 9.8r
7910 = 9.8r

40. EXAMPLE 2
3
7.91×10 = 9.8r
7910 = 9.8r
2
2
7910 = ( 9.8r )

41. EXAMPLE 2
3
7.91×10 = 9.8r
7910 = 9.8r
2
2
7910 = ( 9.8r )
62568100 = 9.8r

42. EXAMPLE 2
3
7.91×10 = 9.8r
7910 = 9.8r
2
2
7910 = ( 9.8r )
62568100 = 9.8r
9.8 9.8

43. EXAMPLE 2
3
7.91×10 = 9.8r
7910 = 9.8r
2
2
7910 = ( 9.8r )
62568100 = 9.8r
9.8 9.8
r = 6384500

44. EXAMPLE 2
3
7.91×10 = 9.8r
7910 = 9.8r
2
2
7910 = ( 9.8r )
62568100 = 9.8r
9.8 9.8
r = 6384500
The radius of Earth is about 6384500 meters.

45. HOMEWORK

46. HOMEWORK
p. 138 #1-51 odd
“If fifty million people say a foolish thing, it is still a foolish
thing.” - Anatole France